I have two systems of coupled integral equations to solve. I know how to solve a single integral equation by reducing it to an eigenvalue problem. How can two systems of coupled integral equations be reduced also to an eigenvalue problem. Can two systems of coupled integral equations be reduced to a single integral equation?
Integral equation can be written in an operator form Ax=x, where x is a function, A - integral operator, see http://www.encyclopediaofmath.org/index.php/Integral_operator , https://en.wikipedia.org/wiki/Integral_equation .
Your question is rather complex and a lot depends on the exact form.
Maybe you can find something useful in this article: http://dea.ele-math.com/04-14/Solving-a-system-of-nonlinear-integral-equations-and-existence-of-asymptotically-stable-solutions
and here: http://www.hindawi.com/journals/aaa/2014/236065/
This article suggests analytical method: http://www.sciencedirect.com/science/article/pii/S0895717711001919
i found the book linear and nonlinear integral equations by AbdulMajid helpful. The book treated solutions of systems of integral equations using Adomian decomposition and other mathematical methods. I will try to adapt the method(s) to my work.
It is hard to say yes or no, depends on the equations. Generally, linear and nonlinear integral eq.s will be helpful, it can give new solutions of the system based on a known solution.
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